Integral Inclusions Research Articles

Solvability of nonlinear impulsive Volterra integral inclusions and functional differential inclusions*1

This paper presents sufficient conditions for the existence of solutions to nonlinear impulsive Volterra integral inclusions and initial value problems for second order impulsive functional differential inclusions in Banach spaces. Our results are obtained via a fixed point theorem due to Hong [J. Math. Anal. Appl. 282 (2003) 151–162] for discontinuous multivalued increasing operators.

Fixed Points for Multi-Valued Mixed Increasing Operators in Ordered Banach Spaces with Applications to Integral Inclusions

Some new fixed point and coupled fixed point theorems for multi-valued mixed increasing operators in ordered Banach spaces are presented in this paper. As applications, we prove the existences of solutions for a class of integral inclusions.

Fixed points of discontinuous multivalued increasing operators in Banach spaces with applications

By using relatively weakly compactness conditions, we obtain existence theorems of fixed points for discontinuous multivalued increasing operators in Banach spaces. As an application, we utilize the results obtained in Section 2 to lead to the existence principles for integral inclusions of Hammerstein type multivalued maps.

Fixed-point theorems for countably condensing maps on Fréchet spaces

New fixed-point results for countably condensing operators defined on Fréchet spaces are presented. Applications to integral inclusions are also discussed.

A generalization of the Petryshyn–Leggett–Williams fixed point theorem with applications to integral inclusions

The existence of three fixed points is established for multivalued maps which satisfy an axiomatic index theory. Our results enable us to develop criteria for the existence of three nonnegative solutions to integral inclusions.

Continuums of solutions for integral inclusions on the half line

In this paper we show that the solution set of certain Volterra inclusions on the half line is a continuum.

On some classes of operator inclusions with lower semicontinuous nonlinearities

We consider a class of multimaps which are the composition of a superposition multioperator ${\mathcal P}_F$ generated by a nonconvex-valued almost lower semicontinuous nonlinearity $F$ and an abstract solution operator $S$. We prove that under some suitable conditions such multimaps are condensing with respect to a special vector-valued measure of noncompactness and construct a topological degree theory for this class of multimaps yielding some fixed point principles. It is shown how abstract results can be applied to semilinear inclusions, inclusions with $m$-accretive operators and time-dependent subdifferentials, nonlinear evolution inclusions and integral inclusions in Banach spaces.

Set-valued integral equations of fractional-orders

The topic of fractional calculus (derivative and integral of arbitrary orders) is enjoying growing interest not only among Mathematicians, but also among physicists and engineers (see [6–16, 18–20, 23–25]). The set-valued integral equations (integral inclusions) arises in the study of control system (see [21, 22, 26]). In this paper we prove the existence of locally bounded variation solution of a Volterra type set-valued integral equation of arbitrary (not necessarily integer) order. The proof will be based on the measure of weak noncompactness and the existence of Caratheodory selectors. As a consequence we study the initial value problem for some set-valued differential and integro-differential equations. The corresponding single-valued problems will be firstly considered.

Fixed Point Theorems for Set-Valued Maps and Existence Principles for Integral Inclusions

New fixed point theorems of Mönch type are presented for set-valued maps. These theorems are then used to establish general existence principles for Hammerstein integral inclusions in Banach spaces.

A Note on the Solution Set of Integral Inclusions

A Note on the Solution Set of Integral Inclusions

A Note on the Existence of Multiple Fixed Points for Multivalued Maps with Applications

In this paper we discuss the existence of multiple fixed points for multivalued maps. Our fixed point results automatically lead to new existence principles for integral inclusions.

Nonlinear alternatives for multivalued maps with applications to operator inclusions in abstract spaces

A nonlinear alternative of Leray–Schauder type is presented for condensing operators with closed graph. We will then use this theorem to establish new existence principles for differential and integral inclusions in Banach spaces.

Topological properties of the set of fixed points of a multivalued map

This paper is a study of the topological properties of the set of fixed points of a multivalued map. Some theorems are proved on the topological dimension of this set, and conditions are studied under which the set is connected or acyclic. The theorems obtained are applied to the investigation of the set of solutions of differential and integral inclusions.

Integral inclusions of upper semi-continuous or lower semi-continuous type

Topological results for set valued maps are used to establish existence results for integral inclusions of Volterra or Hammerstein type.

A Sensitivity Analysis of Volterra Integral Inclusions with Applications to Optimal Control Problems

We consider a class of parametrized Volterra integral inclusions associated to multivalued m-accretive operators in a Banach space. We show that the solution set depends continuously on the parameter, in a specified sense. The continuous dependence results are used to study the variational stability of distributed parameter optimal control problems.

INTEGRAL INCLUSIONS WITH NONCONVEX IMAGES, AND THEIR APPLICATIONS TO BOUNDARY VALUE PROBLEMS FOR DIFFERENTIAL INCLUSIONS

This paper contains a treatment of an integral inclusion of Hammerstein type generated by the product of a linear integral operator and a multivalued mapping with images convex with respect to switching. This product is not a Volterra operator in general. Estimates of the closeness of a solution of the inclusion to a given function are proved on the basis of the theory of existence of continuous branches of multivalued mappings with images convex with respect to switching. By using these estimates it is proved that the solution set of the original inclusion is dense in the solution set of the convexified inclusion in the space of continuous functions. In the case when the kernel of the linear operator consists solely of the zero element the “bang-bang” principle is proved for the Hammerstein inclusion. In the second part of the paper the theory is used for investigating boundary value problems for differential inclusions with nonconvex right-hand side. Bibliography: 34 titles.

On the existence of solutions for Volterra integral inclusions in Banach spaces

In this paper we examine a class of nonlinear integral inclusions defined in a separable Banach space. For this class of inclusions of Volterra type we establish two existence results, one for inclusions with a convex‐valued orientor field and the other for inclusions with nonconvex‐valued orientor field. We present conditions guaranteeing that the multivalued map that represents the right‐hand side of the inclusion is α‐condensing using for the proof of our results a known fixed point theorem for α‐condensing maps.

On integral inclusions of Volterra type in Banach spaces

On integral inclusions of Volterra type in Banach spaces

Existence of solutions for integral inclusions of Urysohn type with nonconvex-valued orientor field

In this paper, we prove the existence of solutions for an integral inclusion of Urysohn type with nonconvex orientor field and with delay. We make standard boundedness and continuity assumptions on the data, and we assume that the orientor field is l.s.c. in the state variable. Using a selection theorem of Fryszkowski, we are able to prove the existence of solutions, extending an earlier result of Angell.

Existence and convergence results for integral inclusions in Banak spaces

Existence and convergence results for integral inclusions in Banak spaces